日本オペレーションズ。リサーチ学会 2005年春季研究発表会
2−『一柑
0軋meⅦra侃cⅡmg五mee『五mgⅡ)es五gmamdOp七五m五za七五①mim
O CommⅧm五ca七五omNe七wo抽sw温仙風五地Åma皿y凱S
中科院数学院*呉 軍 WUJun
OllO7734 甲南大学
岳 五一 YUEWuyi
中科院数学院江 寿陽 WANGShouyang
where∧representstochoose thesmalleronebe−
tweentwocomponents,and+representstochoose thepositivepart・Let rI(b,D)denote the mean profitfunction
withthelincarpcnaltycostasfo1lows:
n(岬=γ上b榊)血川上+∞′(埴
1Int.roductionIn[1】a.nd[2】,thestochastic trafBcengineer− ingproblemshavcbcenstudicd・In[3],WCprC−
Sented an analysisfor theloss rate constraintin
SuChproblemsandshowedtheimpactofthelossrateconstraintonthenetworkperformancebynu−
mericalresults.Inthi$paper,WepreSentananaト
ysisforthcmodclprcscntcdin[3]toderivethc
OptimalbandwidthcapacitywithalinearpenaltyCOSt.We alsoanalyze theriskofnetwork pro飢 Shortfa11byuslngmean−Varianceapproach・
2 SystemModel
A Communication Network(CN)should de−
riveitsrevenuebyserv1ngtra氏cdemandincluding
VOice,PaCkct data,imagc andfu11−mOtion vidco・FbrunitbandwidthcapacityallocatedtotheCN,
anunitcostwillbecharged・Fbrunsatisfied11nit
t.ra庁icdemandwiththelimitationofnetworkband−Width,alinearpenalty cost wi11beaddedinthe Objectivefunction.Theobjectiveofthissystemis
tomaximizetheCNmeanprofit.
Uncertain totaltra侃c demandin t,he CN de−
noted by D > Ois characterized by a random
distributionwiththeprobabilitydensityfunction
f(x)andthecumulativedistributionfunctionF(x).
Let b>Odenote theamount ofbandwidthca−
pacity provisionedin the CN,r denote the unitrevenueoftheCNbyservlngthetrafRcdemand,
Cdenote theunit costfor unitbandwidthcapac−
ity allocatedin the CN,and q denote thelinear
penaltycostfortheunsatis丘edunitdemand.Let P(b≧6D)≧1−∈denotethelossrateconstraint,
andlet Ch。X >O denote the maximalcapacity
t.hat can bca1locatcdin thc CN.Tb avoid unrcal− isticcases,WemakethefollowlngaSSumptions: (1)Systemparametersare:r>q>0,r>c>0. (2)Lossrateconstraintpa,rameteris:0≦6≦1. (3)Confidencelevelis:0≦1−e≦1.
3 0p七imal1Bandwid七hAlloca七五onw五th
Pema丑tyCost Let7r(b,D)denotetherandomprofitfunction byservlngtrafRcdemandinthenetworkwiththe linea.rpenaltycost,namely, 打棒,刀)=γ・(わ∧β)−q岬一切+−Cゐ (1) +00 上 (ェーわ)J(霊)血−Cむ・(2)  ̄q Theobjectivefunctionofthesystemis 口*=ロ(ぁげ)), (3) SubjecttoP(b≧6D)≧1−∈andb≦Cha,:・Ⅰ㌣is theoptimalpro丘tfunction・Next,We analy21e the property of tlle mean
profitfunctionⅢ(b,D).Thefirstorderderivative OfⅢ(b,D)withrespecttobisgivenasfollows: dⅢ(む,β) =(r+q−C)−(γ+q)ダ(町 (4) dむ ThesecondorderderivativeofrI(b,D)withre− SPCCttObisglVCnaSfo1lows: d2Il(♭,か) =−(γ+ヴ)J(わ)≦0. (5) dわ2 Therefore,WeCanSaythatⅢ(b,D)isaconcave functionofb.So,thcoptimalbandwidthcapacity ︵︵ 一1 ダ ∫ Or
守 ),Whereダ ̄1(・)
without constraints is is theinversefunction Wedefinethelossrateastheprobabilitythatthetra氏cdemandcannOt beserved by the CN. Thercforc,thclossratcconstraintisequlValcntto
b∈【6F−1(1−e),+∞).Ifweconsiderthemax−
imalcapacity constraint,the optimalbandwidth
CapaCityfortheCNcanbeglVenby
C
γ+q−
∨紺 ̄1(1−∈) ∧q乃α。(6) γ+q
Where v represents to choose thelarger orle be− tweentwocomponents.
Weconsiderafu11ydistributednetwork,Where
the tra且c demandis a5Sumed tofo1low auniform
distributionas aspccialcxampIc.Wcgivcsomc
nllmericalresults to show theimpact ofpenalty
−272−
ヰ去(q+γ)2れ(2γ2+2q2+血汀−3γC −39C+c2)か(去q2+言qr)わ2+吉和(10)
WiththesamesystemparametersforFig・1,
We glVe SOme numericalresults to show theim− PaCtOfriskaNerSeneSS
.
thcincrca5C Of risk avcrscncss.Ordinate axis of
Fig・2correspondstothepercentagedi鮎renceof
theobjectivefunctionftomtheobjectivefunctionwithout risk. OStOnthcbandwidthcapacity(sccFig・1)・Hor−
1ZOntalaxisq/r ofFig.1corresponds to thein− CreaSe Ofpenalty cost.Ordinate axis ofFig.1
COrreSpOndstothepercentagedi鮎renceoftheop−
timalbandwidthcapacityfromtheoptimalband−
width capacity without penalty. For comparingwiththemodelof[1],示echoosesystemparame− tersfortwocasesasfo1lows:(1)r=7.5,C=1.5, and(2)r=7・5,C=0・5,intheinterval[0,1]・ 亜 凡∵仇 乱〓机 正二n仇 盲8£彗とd∞ =:;…£… 皿 孤 皿岬 仙 狐 几 u点︸リ5﹂電せ艮一〇
十l亡75c=1.5 ↓戸尋.5c=α5
nO 81 02 0.3 0.4 0.5 8¢ 0.7+0月 0,9 1.0 如I∼Co雨 Figurel:Impactonbandwidthcapacity. ThenumericalresultsshowninFig.1,includ一 ngtheresultspresentedin[1],reVealadistinctlmpaCtOfpenaltycost on the CN bandwidthca−
pacity.Withthcsamcpenaltycost,thclargerthe unitcostcis,thegreaterth
Widthcapacityis.
4 RiskAnalysiswithPenalty Cost DuetotheuncertaintrafRcdemand,theprofitisalsouncertain,Wedefinetheriskasthedeviation
fromtheoptimalprofitinthispaper.
Weanalyzctheriskofprofitshorthllbyuslng themean−Varianceapproach.Theobjectivefunc− tion,Whichisdenotedby◎*,isglVenaSfo1lows: ◎*=叩げ)−αⅤ叫咄β)]) (7) Whereα(0≦α≦1)isariskaverseness . (1),andVar[7T(b,D)]isthcvarianccof7T(b,D). Whenthetrafncdemanddist.ributedint,hewhole networkis assumedtofo1lowtheuniformdistribu_tion,WeCanObtainthemeanfunctionby
Ⅲ(岬)=一宇紬伸一C)わー芸・(8)
ThevarianCefunctionisobtainedby叫棉卯=−(q+γ)2れ(2γ2+2q2+c2
埼r−3γC−3qc)か(ヴ2+蝉2+れ9)
Moreover,theobjectivefunction◎■presented inEq・(7)isgivenby 竺◎*=(ト距(叫−C)む−]
0.0 0.10.2 0.3 0.4 0.5 0.8(‖ 0.も 0.9l.O Ris一触lちeneSSP8ralllOter Figurc2:Impactonobjcctivcfunction. ThenumericalresultsshowninFig・2,includ一 ngtheresultspresentedin[1],reVealadistinctlmpaCtOfriskonthcCNobjcctivcfunction・With
thesameriskaverseness,thelargertheunitcostc is,thelesstheimpactontheobjectivefunctionis.5 Conclusion
Inthispaper,WepreSentedastochasticmodel
foroptimizingbandwidtha1locationinCommuni−Cation Networks(CNs)and derived the optimal
bandwidth capacity with thc penalty cost・We
alsoanaly2:edtheriskaversenessinCNsunderthe
mean−Variance framework.N11mericalresult,S・re−Vealedtheimpactsofthepenaltycostandtherisk
aversenessonthenetworkperformance・Acknowledgments
ThisworkwassupportedinpartbyGRANT−
IN−AID FORSCI.RES.(No.16560350) d l ▼▲ a VJ V﹀ b bMEXT.ORC(2004−2008),Japan andin part
NSFCandMADIS,China. Refbrences 【1]D・MitraandQ.Wang,“StochastictrafRcengineer− 1ng,With applicationstonetworkrevenueman− agement,”Proc.L?fIEEE〃膵OCOM,2003・ 【2]D・MitraandQ・Wang,“Risk−aWarCnCtWOrkprofit managementinatwo−tiermarket,”Proc.qf18th